Some Compactness Criteria for Weak Solutions of Time Fractional PDEs

The Aubin--Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the existence of weak solutions to time fractional PDEs. We first define the weak Caputo derivatives of order $\gamma\in (0,1)$ for functions valued in general Banach spaces, consistent with the traditional definition if the space is $\Bbb{R}^d$ and functions are absolutely continuous. Based on a Volterra-type integral form, we establish some time regularity estimates of the functions provided that the weak Caputo derivatives are in certain spaces. The compactness criteria are then established using the time regularity estimates. The existence of weak solutions for a special case of time fractional compressible Navier--Stokes equations with constant density and time fractional Keller--Segel equations in $\Bbb{R}^2$ are then proved as model problems. This work provides a framework for studying weak solutions of nonlinear time fractional PDEs.